Affine projection tensor geometry: Lie derivatives and isometries

TL;DR

This paper extends the generalized projection-tensor geometry to include Lie derivatives and isometries, providing a compact notation and analyzing spacetimes with symmetries such as cosmological models and spherical symmetry.

Contribution

It introduces a compact notation for projected objects and derives fully projected decompositions of Lie derivatives in the context of projection-tensor geometry.

Findings

Derived decompositions of Lie derivatives of projection tensors and metrics.

Applied results to analyze spacetimes with isometries such as cosmological and spherical models.

Provided a unified framework for understanding symmetries in projection-tensor geometry.

Abstract

The generalized projection-tensor geometry introduced in an earlier paper is extended. A compact notation for families of projected objects is introduced and used to summarize the results of the previous paper and obtain fully projected decompositions of Lie derivatives of the projection tensor field, the metric and the projected parts of the metric. These results are applied to the analysis of spacetimes with isometries. The familiar cases of spacetimes with isotropic group orbits --- cosmological models and spherical symmetry --- are discussed as illustrations of the results.